How many stories tall does a building have to be for it to be called a
skyscraper? Most of the definitions I've read just say a skyscraper is
'a very tall building'.

Words have correct uses, not correct definitions. When people use the
word "skyscraper" there is no fixed height at which everyone begins to
use the word to describe a building. I think what you mean to ask is:
Are there any official standards on height for a skyscraper? Your
question sounds very familiar and I seem to remember the answer being
that no official standard exists. I'll leave it up to the researchers
to verify that, though.

You may find these threads interesting:

http://www.hoogbouw.nl/OZforum/cgi-bin/topic.cgi?forum=1&topic=554

http://www.hoogbouw.nl/euroforums/cgi-bin/topic.cgi?forum=1&topic=34

Seven stories is enough (according to Rogers and Hammerstein).

Everything's up to date in Kansas City
They gone about as fer as they can go
They went an' built a skyscraper seven stories high
About as high as a buildin' oughta grow

HI,
You are right, a skyscraper is a very tall building, and that is relative.
The skycrapers of yore in Chicago, NYC, and elsewhere stopped being
considered as such when they lost their prominance in the skyline to
even higher buildings.

It would be interesting to try to figure out how many newer, taller
buildings have to appear before a certain building is no longer
referred to as a skyscraper in that city.   For example, do people
still consider the Seagram's building in NYC to be a skyscraper?

Incidentally, above the height of about 30 stories (say, 120 yards),
at the foot  of the building a person is hardly aware of the
additional height.

Nice answers. Thanks kindly.